TechnischeUniversitatChemnitz-Zwickau DFG-Forschergruppe\SPC"FakultatfurMathematik DomainDecompositionand PreconditioningOperators MultilevelTechniquesfor SergejV.Nepomnyaschikh SiberianBranchofRussianAcademyofSciences e-mail:svnep@comcen.nsk.su ComputingCenter Novosibirsk,63090 SPC9530 Preprint-ReihederChemnitzerDFG-Forschergruppe \ScienticParallelComputing" November1995
1Introduction Inrecentyears,domaindecompositionmethodshavebeenusedextensivelytoeciently 5,15,17].Butdirectrealizationofmultileveltechniquesonaparallelcomputersystem havedevelopedintoaneectivetoolfortheconstructionandanalysisoffastsolvers[2, fortheglobalproblemintheoriginaldomaininvolvesdicultcommunicationproblems. solveboundaryvalueproblemsforpartialdierentialequationsincomplex{formdomains [4,13,16].Ontheotherhand,multileveltechniquesonhierarchicaldatastructuresalso Ithispaper,wepresentandanalyzeacombinationofthesetwoapproaches:domain preconditioningoperators. decompositionandmultileveldecompositiononhierarchicalstructurestodesignoptimal LetR2beapolygon.Inthedomainweconsidertheboundaryvalueproblem 8><>:?2X where i;j=1@ @xiaij(x)@u @N=2X @u@xj+a0(x)u=f(x);x2 i;j=1aij(x)@u @N+(x)u=0; u(x)=0; @xjcos(n;xi) x2?1: x2?0 (1.1) of?0: Weintroducethebilinearforma(u;v)andthelinearfunctionall(v): nitenumberofcurvilinearsegments,?=?0[?1;?0=?0:here?0denotestheclosure istheconormalderivative,ndenotestheoutwardnormalto?,and?0isaunionofa ByH1(;?0)wedenotethesubspaceoftheSobolevspaceH1() a(u;v)=z2x H1(;?0)=nv2H1()jv(x)=0;x2?0o: (1.1)aresuchthatthebilinearforma(u;v)issymmetric,elliptic,andcontinuouson Letussupposethattheoperatorcoecientsandtheright-handsideoftheproblem i;j=1aij(x)@u l(v)=zf(x)vdx: @xj@v @xi+a0(x)uvdx+z?1(x)uvdx H1(;?0)H1(;?0),i.eȧ(u;v)=a(v;u)8u;v2H1(;?0) andthelinearfunctionall(v)iscontinuousonh1(;?0): tionproblem[1]u2h1(;?0):a(u;v)=l(v)8v2h1(;?0): Thegeneralizedsolutionu2H1(;?0)of(1.1)is,bydenition,asolutiontotheprojec- 0kuk2H1()a(u;u)1kuk2H1()8u2H1(;?0) jl(u)jkukh1()8u2h1(;?0): 1 (1.2)
Weknowthatundertheseassumptionsfora(u;v)andl(v)thereexistsauniquesolution of(1.2). whereiarepolygonswithdiametersontheorderofh.letusconsideracoarsegrid triangulationof Letbeaunionofnnonoverlappingsubdomainsi, =n[i=1i;i\j=;;i6=j; andwerenehi;0severaltimes.thisresultsinasequenceofnestedtriangulations h0=n[i=1hi;0;hi;0=m(0) diam((0) hi;0;hi;1;:::;hi;j i;l)=0(h) i[l=1(0) i;l; suchthat wherethetriangles(k+1) subtrianglesbyconnectingthemidpointsoftheedges. IntroducethespacesWi;0Wi;1:::Wi;J=Hh(i); i;laregeneratedbysubdividingtriangles(k) hi;k=m(k) i[l=1(k) i;l;k=0;1;:::;j; Vi;0Vi;1:::Vi;J=Hh(?i); i;lintofourcongruent onthetrianglesinhi;k.thespacevi;kisthespaceoftraceson?ioffunctionsfromwi;k: HerethespaceWi;kconsistsofreal-valuedfunctionswhicharecontinuousonandlinear Vi;k=n'hj'h=uhj?i;withuh2Wi;ko:?i=@i;i=1;2;:::;n: (1.3) whichisanapproximationoftheproblem(1.2). WedenethespaceHh()ofrealcontinuousfunctionswhicharelinearoneachtriangle ofhandvanishat?0: triangulationh.then(1.4)isequivalenttothesystemofmeshequations RNwhosecomponentsarevaluesofthefunctionuhatthecorrespondingnodesofthe Letusconsidertheprojectionproblem Eachfunctionuh2Hh()isputincorrespondencewitharealcolumnvectoru2 uh2hh():a(uh;vh)=l(vh)8vh2hh() Au=f; (1.4) (Au;v)=a(uh;vh)8uh;vh2Hh(); (f;v)=l(vh)8vh2hh(); 2 (1.5)
whereuhandvharetherespectiveinterpolationsofvectorsuandv;(f;v)istheeuclidean scalarproductinrn: Thegoalofthisworkistoconstructasymmetricpositivedenitepreconditioning operatorbfor(1.5)soastosatisfytheinequalities c1(bu;u)(au;u)c2(bu;u) (1.6) wherethepositiveconstantsc1andc2areindependentofhandh,themultiplicationof avectorbyb?1shouldbeeasytoimplement. UsingacombinationofAdditiveSchwarzandFictitiousSpaceMethods,optimalpreconditioningoperatorshavebeenconstructedin[11,12,13]forthecaseofarbitrary (unstructured)grids.however,thatconstructioninvolvesexplicitextensionoperators whoseimplementationforthreedimensionalproblemsisoptimalfromthearithmeticcost andtheconditionnumberpointsofviewbutdicultforpracticerealization.themain goalofthisworkistoconstruct,usingthehierarchicalstructure(1.3),arobustoptimal preconditioningoperator.oneofthecrucialpointsin[11,12,13]andthispaperisusingof non{exactsolversinsubdomainsandexplicitextensionoperators.itmeans,toconstruct optimalpreconditioningoperators,wecandesignnormpreservingoperatorsoffunctions givenat?iintoiwiththeoptimalarithmeticcost(anumberofarithmeticoperations shouldbeproportionaltoanumberdegreesoffreedom)andthen,insteadofexactsolvers insubdomains,wecanuseanyspectrallyequivalentpreconditioningoperators.optimal extensionoperatorshavebeenpresentedin[8,9,11]forunstructuredgridsandrobust explicitextensionoperatorsonhierarchicaldatastructuresin[5,14]. Thepaperisorganizedasfollows.IntheSection2,usingAdditiveSchwarzMethod, wedescribegeneralconstructionofapreconditioningoperatorwithlocalmultilevelpreconditioningoperators.inthesection3,wepresentanoptimalmultilevelextensionof gridfunctionsfromboundariessubdomainsintoinsidesubdomains.inthesection4, weproposeanoptimalinterfacepreconditioningoperatorattheboundariesofthesubdomainswhichinvolvesamultileveldecompositionandcorrespondingexplicitextension operatorsatinterfaces. 2Domaindecomposition{additiveSchwarz-Method Todesignthepreconditioningoperatorforthesystem(1.5),weusetheadditiveSchwarz{ Method[7]andrealizethemainideaoftheconstructionofpreconditionersfrom[13]for thehierarchicalgrids.denotebyhh(i)thesubspaceofhh(i) Hh(i)=nuh2Hh(i)juh(x)=0;x2?io anddenethelocalpreconditioningoperatorsbisuchthat Bi:Hh(i)!Hh(i); c3kuhk2h1(i)(biu;u)c4kuhk2h1(i)8uh2hh(i); wherec3;c4areindependentofhandh.wehereafterusethesamedesignationforan operatoranditsmatrixrepresentation.forinstance,todenebi,wecanusetheso-called BPX{preconditioners[3].Todoit,denotebyff(k) lgnodalbasisfunctionsfromthek{th levelanddene B?1 iuh=jxk=0x f(k) l2hh(i)(uh;f(k) l)l2(i)f(k) l: (2.1) 3
suchthat Letusassumethatwecandenetheextensionoperatorsti uh(x)='h(x);x2?i; ti'h=uh; ti:vi;j?!wi;j withc5independentofhandh.herek'hkh1=2(?i)isthenorm[10]inthesobolevspace H1=2(?i)k'hk2H1=2(?i)=HZ?i('h(x))2dx+Z?iZ?i('h(x)?'h(y))2 kti'hkh1(i)c5k'hkh1=2(?i)8'h2vi;j; jx?yj2dxdy: (2.2) Then,wecandenetheextensionoperatort andforany'h2hh(s) wherehh(s)isthespaceoftracesoffunctionsfromhh()ats t:hh(s)!hh(); uh(x)='h(x);x2s; t'h=uh; S=n[i=1?i Here Theoperatortifrom(2.2)isconstructedintheSection3. Letsatisestothefollowinginequalities kt'hkh1()c5k'hkh1=2(s): c6k'hk2h1=2(s)(';')c7k'hk2h1=2(s)8'h2hh(s); k'hk2h1=2(s)=nxi=1k'hk2h1=2(?i): ditioningoperatorbasfollows wherec6;c7independentofhandh.then,accordingto[11],wecandenetheprecon- B?1=2640B?1 1...B?1 n375+t?1t: (2.3) Here0isthenull-matrixwhichcorrespondstonodesofthetriangulationhatSandBi isfrom(2.1). Thefollowingtheoremisvalid (2.4) independentofhandh. Theorem2.1IftheoperatorBisfrom(2.4),thentheconstantsc1;c2from(1.6)are 4
Themaingoalofthissectionistoconstructtherobustoperatortifrom(2.2).During thissection,weomitthesubscripti. 3Multilevelexplicitextensionoperators wefollowto[5,14].denoteby'(k) Todesigntheextensionoperatort:VJ!WJ; thel2-orthoprojectionfroml2(?)onto(k) itheone-dimensionalsubspacespannedbythisfunction'(k) ~Qk=Nk i;i=1;2;:::;nk,thenodalbasisofvkanddenoteby Xi=1Q(k) i:l2(?)!(k) i;k=0;1;:::;j?1: ianddenote i i:dene any'h2vjk'h0k2h1=2(?)+1hk'h1k2l2(?)+j'h1j2h1=2(?)c8k'hk2h1=2(?); Thefollowinglemmasarevalid[14]. Lemma3.1Thereexistsapositiveconstantc8,independentofhandH,suchthatfor Fork=Jwedene~QJastheL2-orthoprojectionfromL2(?)ontoVj. Lemma3.2Thereexistsapositiveconstantc9,independentofhandH,suchthat where Here j'hj2h1=2(?)=z?z?('h(x)?'h(y))2 'h0=~q0'h;'h1='h?'h0: jx?yj2dxdy: (3.1) Denotebyx(k) where'h0;'h1from(3.1). TheconstructionoftheoperatortisbasedonthedecompositionfromtheLemma3.2. k'h0k2+1hk~q0'h1k2l2(?)+jxk=12kk(~qk?~qk?1)'h1k2l2(?)c9k'hk2h1=2(?); thefollowingway.forany'h2vjset iareenumeratedrston?andtheninside)anddenetheextensionoperatortin i;i=1;2;:::;lk,thenodesofthetriangulationhk(weassumethatnodes Then hk=(~qk?~qk?1)'h;k=1;2;:::;j: h0=~q0'h; 'h= h0+ h1+:::+ hj: (3.2) 5
Denetheextensionuhk2Wkasfollows uh0(x(0) ; h0(x(0) i);x(0) Hereis,forinstance,themeanvalueofthefunction uhk(x(k) i)=8<: k=1;2;:::;j: 0; hk(x(k) i);x(k) i=2?; i2?; i62?; h0on? (3.3) =1N0N0 Xi=1 0;1;:::;J?1:Butinthiscasethecostofthedecomposition(3.2)isexpensive(especially Dene Remark3.1WecanusetheL2-orthoprojectionsformL2(?)ontoVkinsteadof~Qk;k= t'h=uhuh0+uh1+:::+uhj h0(x(0) i): Theorem3.1Thereexistsapositiveconstantc10,independentofhandH,suchthat forthreedimensionalproblems). (3.4) Heretheoperatortisfrom(3.2){(3.4). Remark3.2Itisobviousthat kt'hkh1()c10k'hkh1=2(?)8'h2vj: andthecostoftheactionoftandtisproportionaltothenumberofnodesofthegrid domain. Q(k) i'h=('h;'(k) 4Interfacepreconditioningoperators ('(k) i;'(k) i)l2(?)'(k) i [13].LetSbeaunionofKnonoverlappingedgesEiofthetriangulationh0 satises(2.3).todoit,weusetheideaofadditiveschwarzmethodatinterfacesfrom Inthissection,weconstructanoptimalinterfacepreconditionerinthespaceHh(S)which SplitHh(S)intoavectorsumofsubspaces S=K[ whereu0isthecoarsespacewhichconsistsofcontinuousfunctionslinearontheedges Hh(S)=U0+U1+:::+UK; j=1ej;ej\ei=;;i6=j: Ej,j=1;2;:::;K,andUj,j=1;2;:::;K,correspondtoEjandaredenedbelow. 6 (4.1)
ForanyedgeEjwedenetheexplicitextensionoperatorj Denoteby Uj=f'h2Hh(S)j'h(x)=0;x62Ejg; ~U(k) aredierfromthefunctions'(k) asfollows.denoteby'(k) j=vkjej;k=0;1;:::;j: denoteby(k) j;i,i=1;2;:::;i(k) j:~u(j) j!hh(s) correspondingl2-orthoprojection.set j;itheone-dimensionalsubspacespannedbythisfunction'(k) ifromthesection3onlyattheendpointsofej)and j;i:l2(ej)!(k) j,thenodalbasisof~u(k) j(thefunctions'(k) j;i:denej;i extensionoperatorjaccordingto(3.2){(3.4).forany'h2~u(j) anddene~q(k) JastheL2-orthoprojectionfromL2(Ej)onto~U(J) ~Q(k) j=i(k) jxi=1q(k) j;i;k=0;1;:::;j?1; j:nowwecandenethe and hk=(~q(k) h0=~q(0) j'h; j?~q(k?1) j)'h;k=1;2;:::;j jset uhk=8><>: 0; hk(x(k) i);x(k) i2ej i62ej;k=0;1;:::;j (4.3) (4.2) theorem. Dene ThenfromtheTheorem3.1and[13]forthedecomposition(4.1)wehavethefollowing j'h=uh0+uh1+:::+uhj: Theorem4.1Thereexistsapositiveconstantc11;dependentofhandH,suchthatfor anyfunction'h2hh(s)thereexist'hj2uj,j=0;1;:::;k,suchthat Uj=Uj+j~Uj: Lettheoperator0generatesanequivalentnorminU0: k'h0k2h1=2(s)+k'h1k2h1=2(s)+:::+k'hkk2h1=2(s)c11k'hk2h1=2(s) c12k'hk2h1=2(s)(0';')c13k'hk2h1=2(s)8'h2u0; 'h0+'h1+:::+'hk='h; 7 (4.4)
DenotebyjandejtheBPX-likepreconditionersinthespacesUjand~Uj,respectively wherec12;c13independentofhandh.denelocalpreconditionersforuj,j=1;2;:::;k. Then,denetheinterfacepreconditioningoperatorinthefollowingway?1 i'h=jxk=0x supp'(k) j;iej('h;'(k) j;i\ej6=;('h;'(k) j;i)l2(ej)'(k) j;i8'h2uj;?1=+0+kxj=1(?1 j+je?1 jj): j;i8'h2~uj: operator?1 Here+0isapseudo-inverseof0from(4.4),jisfrom(4.2),(4.3),andweextendthe independentofhandh. Theorem4.2Iftheoperatorisfrom(4.5)thentheconstantsc6;c7form(2.3)are jbyzerooutsideej.thefollowingtheoremisvalid. (4.5) dimensionalproblems. Remark4.2Usingcombinationofpresentedtechniqueandtechniquefrom[10],eective Remark4.1Themethodsuggestedinthispapercanbegeneralizedevidentlyforthree TheauthorwishtoacknowledgetheDFG{Forschergruppe\SPC"ChemnitzandtheISF forsupportingthisresearchundergrantsla767/3andnpb000. preconditioningoperatorsforellipticproblemswithjumpcoecientscanbeconstructed. References Acknowledgment [1]J.-P.Aubin.Approximationofellipticboundary{valueproblems.Wiley{Interscience, [2]J.H.Bramble.MultigridMethods.ResearchNotesinMathematicsSeries.Pitman, NewYork,London,Sydney,Toronto,1972. [4]M.DryjaandO.B.Widlund.Multileveladditivemethodsforellipticniteelementproblems.InW.Hackbusch,editor,ParallelAlgorithmsforPartialDierential Comput.,55(191):1{22,1990. Boston-London-Melbourne,1993. [3]J.H.Bramble,J.E.Pasciak,andJ.Xu.Parallelmultilevelpreconditioners.Math. Equations,pages58{69,Braunschweig,1991.Vieweg{Verlag.Proc.oftheSixth [5]G.Haase,U.Langer,A.Meyer,andS.V.Nepomnyaschikh.Hierarchicalextension GAMM{Seminar,Kiel,January19{21,1990. andlocalmultigridmethodsindomaindecompositionpreconditioners.east{west J.Numer.Math.,2(3):173{193,1994.8
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